On rankings and top choices in random utility models with dependent utilities

Different aggregate preference orders based on rankings and top choices have been defined in the literature to describe preferences among items in a fixed set of alternatives. A useful tool in this framework is constituted by random utility models, where the utility of each alternative, or object, is represented by a random variable, indexed by the object, which, for example, can capture the variability of preferences over a population. Applications are derived in diverse research fields, including computer science, management science and reliability. Recently, some stochastic ordering conditions have been provided for comparing alternatives by means of some aggregate preference orders in the case of independent random utility variables by Joe (Math Soc Sci 43:391–404, 2002). In this paper we provide new conditions, based on some joint stochastic orderings, for aggregate preference orders among the alternatives in the case of dependent random utilities. We also provide some examples of application in different research fields.      

Continuous extension of preferences

We consider the problem of extending preferences from a subset of a commodity space to the entire space. It is a simple consequence of the Tietze extension theorem that continuous preferences can be extended if they are defined on closed subsets of a normal space and are representable by utility functions. We show the following: If the space is a non-separable metric space, then extension of preferences is not always possible. In fact for (path-connected) metric spaces, extension property, utility representation property, and separability are equivalent to each other.     

Continuous representation of Von Neumann–Morgenstern preferences

This paper considers a continuous representation of preference relations satisfying Grandmont's (1972) Expected Utility Hypothesis. We equip the preferences with the topology of closed convergence, then we show the existence of a jointly continuous expected utility function and consider its uniqueness. Furthermore, we construct an embedding map of the preferences into the set of expected utility functions.     

A von Neumann–Morgenstern representation result without weak continuity assumption

In the paradigm of von Neumann and Morgenstern (1947), a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on an interval of the real line, it is shown that any affine preference order which is monotone with respect to the first stochastic order admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine preference order on the subset of lotteries with compact support, which is monotone with respect to the second stochastic order, can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference orders on the subset of those lotteries which fulfill some integrability conditions. The subtleties of the weak topology are illustrated by some examples.     

On aggregation and welfare analysis

In this article, the beginnings of a new approach to the theory of aggregation are developed. The basic idea is that aggregation should involve two things: (a) data over which social preferences are defined should be mapped into a smaller-dimensional space, and (b) there should exist an ordering on that lower-dimensional space such that an improvement in this criterion implies an improvement for any of a class of social preference functions and of a class of individual preference relations defined over the original space. Results are developed which show that this conception of aggregation can yield meaningful results; particularly with respect to comparisons of ‘real national income’ in two situations, for a given economy.     

The geometry of revealed preference

In this paper, we examine how the geometry underlying revealed preference determines the set of preferences that can be revealed by choices. Specifically, given an arbitrary binary relation defined on a finite set, we ask if and when there exists a data set which can generate the given relation through revealed preference. We show that the dimension of the consumption space affects the set of revealed preference relations. If the consumption space has more goods than observations, any revealed preference relation can arise. Conversely, if the consumption space has low dimension relative to the number of observations, then there exist both rational and irrational preference relations that can never be revealed by choices.     

Preference symmetries,partial differential equations,and functional forms for utility

A discrete symmetry of a preference relation is a mapping from the domain of choice to itself under which preference comparisons are invariant; a continuous symmetry is a one-parameter family of such transformations that includes the identity; and a symmetry field is a vector field whose trajectories generate a continuous symmetry. Any continuous symmetry of a preference relation implies that its representations satisfy a system of PDEs. Conversely the system implies the continuous symmetry if the latter is generated by a field. Moreover, solving the PDEs yields the functional form for utility equivalent to the symmetry. This framework is shown to encompass a variety of representation theorems related to univariate separability, multivariate separability, and homogeneity, including the cases of Cobb–Douglas and CES utility.     

Ordering utility functions based on mean-seeking behavior1

We consider utility function partial orderings to predict comparative portfolio features with two risky assets. No utility ordering can predict comparative holdings of the riskier asset for any reasonable definition of the latter. We thus reinterpret results of Arrow-Pratt and Ross as predicting comparative mean-seeking behavior. We also present a stronger utility ordering which predicts comparative portfolio means with no joint distribution restrictions. Thus we present a progression of contexts, with successively more relaxed distributional restrictions and hence successively stronger restrictions on utility function pairs, in which comparative mean-seeking behavior (not comparative risk avoidance) is predictable.     

Interview design for revealing preferences of policy makers

A policy maker is asked a few simple questions about his preference. Then the model represents it by a quadratic utility function, which can be made monotonic and quasi-concave (= to provide the convexity of the preference). The design of the interview with a policy maker is aimed at attaining the following goals: (a) no ambiguous output (= degeneration of the model), (b) ordinal approach to preferences (= asking questions about ordinal preferences and providing the uniqueness of the ordinal preference at the model output, regardless of its representation by a quadratic utility function), (c) stability of the model (= the model's input–output transformation is continuous). We also describe briefly the implementation of our model in a user-friendly interface to a corresponding computer program.     

A utility representation theorem with weaker continuity condition

We prove that a mixture continuous preference relation has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg (1941) and Debreu (1959, 1964), but our condition on the continuity of a preference relation is strictly weaker than the usual continuity assumed by them.     

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.     

From revealed preference to preference revelation

Utility functions are regarded as elements of a linear space that is paired with a dual representation of choices to demonstrate the similarity between preference revelation and the duality of prices and quantities in revealed preference. With respect to preference revelation, quasilinear versus ordinal utility and choices in an abstract set versus choices in a linear space are distinguished and their separate and common features are explored. The central thread uniting the various strands is the subdifferentiability of convex functions.     

Concavifiability and constructions of concave utility functions

Concavifiable convex preference orderings are characterized and minimally concave utilities are constructed, using three different approaches. One involves the intersection of arbitrary lines with the three indifference surfaces, another involves conditions on the normals of two indifference surfaces and is related to the super-gradient map of a possible concave utility. In the third approach it is assumed that the ordering is induced by a twice-differentiable utility and Perror's integral of a certain expression formed from the derivatives is used. A possible economic interpretation of minimally concave utilities is suggested, and it is shown that one cannot select concave utilities so that they depend continuously on the ordering.     

Proper preferences and quasi-concave utility functions

It is shown that preferences which are continuous, convex and uniformly proper [Mas-Colell (1983)] on the positive cone of a Banach lattice can be represented by a quasi-concave utility function which is defined on a larger domain with non-empty interior. This utility function may be chosen to be either upper or lower semi-continuous on its domain, and continuous at each point of the positive cone. Conversely, any preference relation on the positive cone which is monotone and arises from such a utility function is shown to satisfy a condition which is slightly weaker than uniform properness but which (in the presence of appropriate compactness assumptions) is sufficient to establish the existence of quasi-equilibria. An example is presented to illuminate the role played by the uniformity requirement.     

Generalized time-invariant overtaking

We present a new version of the overtaking criterion, which we call generalized time-invariant overtaking. The generalized time-invariant overtaking criterion (on the space of infinite utility streams) is defined by extending proliferating sequences of complete and transitive binary relations defined on finite dimensional spaces. The paper presents a general approach that can be specialized to at least two, extensively researched examples, the utilitarian and the leximin orderings on a finite dimensional Euclidean space.     

Demand properties of concavifiable preferences

The purpose of the present paper is to clarify the relation between choice theory for individual consumers, i.e., the observed demand behavior, and the preference ordering ?$?$ of that individual. Specifically, we study how concavifiability (i.e., representability of ?$?$ by a concave utility function) is expressed by quantities (cross-coefficients) appearing in revealed preferences theory. We present a sequence of rather explicit necessary conditions for concavifiability. All these conditions are quantitative asymptotic strengthenings of the strong axiom of revealed preference. The results and concepts are illustrated by means of examples in which an expenditure data is defined by providing its generating utility function.     

Essential supremum and essential maximum with respect to random preference relations

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X$X$ endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X$X$ endowed by a random preference relation to the space L⁰(X)$L^0\left(X\right)$ of X$X$-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.     

On the multi-utility representation of preference relations

We develop the ordinal theory of (semi)continuous multi-utility representation for incomplete preference relations. We investigate the cases in which the representing sets of utility functions are either arbitrary or finite, and those cases in which the maps contained in these sets are required to be (semi)continuous. With the exception of the case where the representing set is required to be finite, we find that the requirements of such representations are surprisingly weak, pointing to a wide range of applicability of the representation theorems reported here. Some applications to decision theory under uncertainty and consumer theory are also considered.     

Expected utility without bounds—A simple proof

We provide a simple proof for the existence of an expected utility representation of a preference relation with an unbounded and continuous utility function.     

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.